Answer:
Answer: K = 4k and H=<u>(2+2k) </u>= k+1
2
Step-by-step explanation:
Part(a) :Let's parametrize the given surface-
X (u, v) = ( u²,v²,u²+kv² )
Now : Xu = (1,0,2u) and Xv = (0,1,2kv)
using below formula:
N (u, v) = <u>Xu x X
v</u>
[ Xu x Xv]
N(u,v) = <u>(-2u,-2kv , 1)</u>
√ {4u²+4k²v²+1}
Now value of normal vector at p=(0,0,0) is N(p)
N(p) = N(0,0 )= (0,0,1)
so finally at p=(0,0,0) : Xu = (1,0,0) , Xv = (0,1,0), Np = (0,0,1)
Clearly Tp(S) is spanned by (1,0,0),(0,1,0)
dN.Xu = d/dt N(t,0)/ t=0 = (-2,0,0)
dN.Xv = d/dr N(0,r)/r=0=(0,-2k,0)
since Xu ,Xv ∈Tp(S)
aXu + bXv ∈Tp(S)
-dNp (aXu + bXv) =(2a,2kb,0)
and represent above in matrix form:
2 0 a
0 2k b
-dNp is symmetric matrix with eigen values (LAMDA) ∧1=2 and ∧2 =2k
Part (b):
Product of Eigen Values is called Gaussian Curvature(K).
Mean of Eigen Values is called Mean Curvature(H).
Therefore K = 4k and H=<u>(2+2k) </u>= k+1
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